How to summarise overlaps between data points

I have a data set of animals passing an RFID reader, it looks like this -

ID    date_time                 
A     2019-11-02 08:07:47    
B     2019-11-02 08:07:48 
A     2019-11-02 08:07:49
A     2019-11-02 08:07:50
A     2019-11-02 08:09:12
A     2019-11-02 08:09:13
B     2019-11-02 08:09:17

I asked this question recently, (combine multiple rows into one time interval), and now my data looks like this - (with the data organised into ten second intervals)

ID     start_date_time.      end_date_time
A      2019-11-02 08:07:47   2019-11-02 08:07:50
B      2019-11-02 08:07:48   2019-11-02 08:07:48
A      2019-11-02 08:09:12   2019-11-02 08:09:13
B      2019-11-02 08:09:17   2019-11-02 08:09:47

I have also added a column which summarises the intervals

dat$Interval = interval(dat$start_date_time,dat$end_date_time)

I now need to find and summarise where these intervals intersect and produce this as a count, to show the number of times animals interact (or are present at the RFID reader at the same time) something like this - (and without repeating reverse interactions, i.e. A-B, B-A)

ID     ID2     Interactions(n)
A      A       0
A      B       1    
A      C       3

Any help appreciated.